Lecture Control system design: Frequency response methods presents the following content: Frequency response plots, performance specifications in the frequency domain, frequency response methods using control system software.
Nguyễn Công Phương CONTROL SYSTEM DESIGN Frequency Response Methods Contents I Introduction II Mathematical Models of Systems III State Variable Models IV Feedback Control System Characteristics V The Performance of Feedback Control Systems VI The Stability of Linear Feedback Systems VII The Root Locus Method VIII.Frequency Response Methods IX Stability in the Frequency Domain X The Design of Feedback Control Systems XI The Design of State Variable Feedback Systems XII Robust Control Systems XIII.Digital Control Systems s i tes.google.com/site/ncpdhbkhn Frequency Response Methods Introduction Frequency Response Plots Performance Specifications in the Frequency Domain Frequency Response Methods Using Control System Software s i tes.google.com/site/ncpdhbkhn Introduction (1) • The frequency response of a system: the steady – state response of the system to a sinusoidal input signal • The sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady state • It differs from the input waveform only in amplitude and phase angle s i tes.google.com/site/ncpdhbkhn Introduction (2) T ( s) = m (s ) = q( s ) m (s ) n ∏ (s + p ) i i =1 r(t ) = Asin ωt → R(s ) = → Y (s) = R (s )T ( s) = Aω s2 + ω k1 k αs + β + + n + s + p1 s + pn s + ω αs + β → y (t ) = k1e− p1t + + kne − pnt + L−1 2 s +ω αs + β ysteady − state (t ) = lim y(t ) = lim L−1 2 t →∞ t →∞ s +ω = A T ( jω ) sin(ω t + φ ), φ = ∠T ( jω ) s i tes.google.com/site/ncpdhbkhn Frequency Response Methods Introduction Frequency Response Plots Performance Specifications in the Frequency Domain Frequency Response Methods Using Control System Software s i tes.google.com/site/ncpdhbkhn Frequency Response Plots (1) G ( jω ) = G ( s ) s = jω = Re[G ( jω )] + Im[G ( jω )] = R(ω ) + jX (ω ) = G ( jω ) e jφ (ω ) = G ( jω ) ∠[φ (ω )] = R (ω ) + X (ω )∠{tan −1 [ X (ω ) / R (ω )]} s i tes.google.com/site/ncpdhbkhn Frequency Response Plots (2) Ex 1 V1( s) I (s) = Cs Cs + R Cs V2 (s ) → G( s) = = V1 (s ) RCs + + V2 (s ) = Vcapacitor (s ) = = = = j ( ω / ω1 ) + V1 ( s ) jω ( RC ) + , 1 + (ω / ω1 ) ∠ tan (−ω / ω1 ) −1 C − Positive ω 0.5 Negative ω 0.4 ω1 = RC ω / ω1 − j + (ω / ω1) + (ω / ω1 ) V2 ( s ) − 0.3 0.2 0.1 X(ω) → G( jω ) = G ( s ) s = jω = + R -0.1 -0.2 -0.3 -0.4 -0.5 0.2 0.4 0.6 0.8 R(ω) s i tes.google.com/site/ncpdhbkhn Frequency Response Plots (3) Semilog plots of the magnitude (in decibels) and phase (in degrees) of a transfer function versus frequency TdB = 20log10 T T =T φ → φ ( )( )( ) T = T1T2 T3 = T1 φ1 T2 φ2 T3 φ3 = ( T1T2T3 ) φ1 + φ2 + φ3 + 20 log10 T = 20log10 T1 + 20log10 T2 + 20log10 T3 + → φ = φ1 + φ2 + φ3 + s i tes.google.com/site/ncpdhbkhn Frequency Response Plots (4) 2 j ω j ζ ω j ω + K ( jω) ±1 1 + 1 + z1 ωk ωk T(ω) = jω j2ζ 2ω jω + 1 + 1 + p1 ωn ωn K : gain : pole at the origin jω jω : zero at the origin jω 1+ p1 : simple pole 1+ j 2ζ 2ω ωn jω + ω n : quadratic pole jω 1+ : simple zero + j2ζ 1ω + jω : quadratic zero z1 ωk ωk s i tes.google.com/site/ncpdhbkhn 10 Frequency Response Plots (32) Ex T(ω ) = 1000( jω + 20) 8(1 + jω / 20) = jω(1 + jω / 5)2 1 + jω /10 + ( jω /10)2 jω ( jω + 5) ( jω ) + 40 jω + 100 40 TdB 20 log 10 20 log10 20 (1 + jω / 5)2 20 log 10 + ω 20 log10 j 2ζ ω jω 2 1 + + ω k ωk N -20 jω 20 jω -40 ωk ω -60 −40 N dB/decade -80 20 log 10 -100 0.1 1 + jω4 /10 + ( jω /10) s i tes.google.com/site/ncpdhbkhn 10 20 50 100 38 Frequency Response Plots (33) Ex T(ω ) = 1000( jω + 20) 8(1 + jω / 20) = jω(1 + jω / 5)2 1 + jω /10 + ( jω /10)2 jω ( jω + 5) ( jω ) + 40 jω + 100 40 TdB 20 ω -20 -40 -60 -80 -100 0.1 Frequency Response s i tes.google.com/site/ncpdhbkhn 10 20 50 100 39 Frequency Response Plots (34) Ex 1000( jω + 20) 8(1 + jω / 20) = jω(1 + jω / 5)2 1 + jω /10 + ( jω /10)2 jω ( jω + 5) ( jω ) + 40 jω + 100 T(ω ) = φ 90 50 ω -50 1 + jω z N -90 90N o -200 0o z 10 -180 z 10z ω -270 -300 -360 0.1 0.2 0.5 s i tes.google.com/site/ncpdhbkhn 10 50 100 200 40 500 Frequency Response Plots (35) Ex T(ω ) = 1000( jω + 20) 8(1 + jω / 20) = jω(1 + jω / 5)2 1 + jω /10 + ( jω /10)2 jω ( jω + 5) ( jω ) + 40 jω + 100 φ 90 50 ω -50 ( jω ) N -90 -180 ω −90N o -200 -270 -300 -360 0.1 0.2 0.5 s i tes.google.com/site/ncpdhbkhn 10 50 100 200 41 500 Frequency Response Plots (36) Ex T(ω ) = 1000( jω + 20) 8(1 + jω / 20) = jω(1 + jω / 5)2 1 + jω /10 + ( jω /10)2 jω ( jω + 5) ( jω ) + 40 jω + 100 φ 90 50 ω -50 jω 1+ p p 10 o p -90 N -180 10p -200 ω −90N o -270 -300 -360 0.1 0.2 0.5 s i tes.google.com/site/ncpdhbkhn 10 50 100 200 42 500 Frequency Response Plots (37) Ex T(ω ) = 1000( jω + 20) 8(1 + jω / 20) = jω(1 + jω / 5)2 1 + jω /10 + ( jω /10)2 jω ( jω + 5) ( jω ) + 40 jω + 100 φ 90 50 ω -50 j 2ζ ω jω 2 1 + + ωk ωk ωk 10 0o ωk N 10ωk -90 -180 ω − 180N o -200 -270 -300 -360 0.1 0.2 0.5 s i tes.google.com/site/ncpdhbkhn 10 50 100 200 43 500 Frequency Response Plots (38) Ex Find the transfer function from the Bode plot? 40 ( jω) N H dB Sl ope = 30 20 20N dB/decade ω 10 jω ω -10 -20 0.1 s i tes.google.com/site/ncpdhbkhn 10 20 100 200 44 500 Frequency Response Plots (39) Ex Find the transfer function from the Bode plot? 40 K H dB Sl ope = 20 log10 10 30 20 20 log10 K ω 10 jω ω -10 -20 0.1 s i tes.google.com/site/ncpdhbkhn 10 20 100 200 45 500 Frequency Response Plots (40) Ex Find the transfer function from the Bode plot? 40 Sl ope = 20 log10 10 30 N jω 1 + p 20 p ω −20N dB/decade H dB 10 jω ω 20log10 -10 -20 0.1 1 + jω / 5 s i tes.google.com/site/ncpdhbkhn 10 20 100 200 46 500 Frequency Response Plots (41) Ex T(ω ) = Find the transfer function from the Bode plot? 40 jω 1+ p Sl ope = 20 log10 10 30 N 20 p ω −20N dB/decade H dB 10 jω (1 + jω / 5)(1 + jω /10) 10 jω 20 log10 1 + jω /10 ω 20log10 -10 -20 0.1 1 + jω / 5 s i tes.google.com/site/ncpdhbkhn 10 20 100 200 47 500 Frequency Response Methods Introduction Frequency Response Plots Performance Specifications in the Frequency Domain Frequency Response Methods Using Control System Software s i tes.google.com/site/ncpdhbkhn 48 Performance Specifications in the Frequency Domain (1) ωresonant = ωn − 2ζ , ζ < 0.707 ( M pω = G ( jωr ) = 2ζ − 2ζ 20 G (ω ) = ) −1 , ζ < 0.707 ζ = 0.05 TdB 10 ζ = 0.2 ζ = 0.4 j2ζ 2ω jω 1+ + ωn ω n -10 -20 ζ = 0.707 ζ = 1.5 -30 -40 ω 0.1ω n s i tes.google.com/site/ncpdhbkhn ωn 10ω n 100ω n 49 Performance Specifications in the Frequency Domain (2) ( M pω = G( jωr ) = 2ζ − 2ζ 5.5 ) −1 , ζ < 0.707 4.5 3.5 M pω 2.5 1.5 0.1 0.2 0.3 0.4 ζ 0.5 0.6 0.7 y( t ) = + Be −ζωn t cos(ω1t + θ ) Resonant magnitudes should be relatively small: Mpω < 1.5, for example s i tes.google.com/site/ncpdhbkhn 50 Frequency Response Methods Introduction Frequency Response Plots Performance Specifications in the Frequency Domain Frequency Response Methods Using Control System Software s i tes.google.com/site/ncpdhbkhn 51 Ex T (s ) = Frequency Response Methods Using Control System Software 500s (s + 5)( s + 10) s i tes.google.com/site/ncpdhbkhn 52 ... ω 1+ = tan φ − p1 p1 φ 0.1p1 p1 -1 0 10 p1 100 p1 ω -2 0 -3 0 -4 0 -5 0 -6 0 -7 0 -8 0 -9 0 -1 00 s i tes.google.com/site/ncpdhbkhn 15 Frequency Response Plots (10) TdB jω TdB = 20log10... tes.google.com/site/ncpdhbkhn Frequency Response Methods Introduction Frequency Response Plots Performance Specifications in the Frequency Domain Frequency Response Methods Using Control System Software s... XIII.Digital Control Systems s i tes.google.com/site/ncpdhbkhn Frequency Response Methods Introduction Frequency Response Plots Performance Specifications in the Frequency Domain Frequency Response Methods